Ray Long wants to retire in Arizona when he is 70 years of age. Ray is now 50. He believes he will need $130,000 to retire comfortably. To date, Ray has set aside no retirement money. Assume Ray gets 14% interest compounded semiannually. How much must Ray invest today to meet his $130,000 goal

To find out how much Ray needs to invest today to meet his $130,000 goal, we can use the formula for compound interest, which is:

A = P(1 + (r/n))^(nt)

Where:
A = the future value (amount Ray wants to have at retirement)
P = the principal (the initial amount Ray needs to invest today)
r = the annual interest rate (14%, or 0.14 as a decimal)
n = the number of times interest is compounded per year (semiannually, so n = 2)
t = the number of years (70 - 50 = 20)

Plugging in the values, we get:

130,000 = P(1 + (0.14/2))^(2 * 20)

Now, let's solve for P:

130,000 = P(1 + 0.07)^40

130,000 = P(1.07)^40

Divide both sides by (1.07)^40:

P = 130,000 / (1.07)^40

Using a calculator, we find:

P ≈ $18,587.78

So, Ray needs to invest approximately $18,587.78 today to meet his $130,000 retirement goal.

To determine how much Ray must invest today to meet his $130,000 retirement goal, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment (in this case, $130,000)
P = the principal amount (the initial investment)
r = the annual interest rate (14% or 0.14)
n = the number of times interest is compounded per year (semiannually, meaning twice a year)
t = the number of years the money is invested for (20 years, since Ray is currently 50 and plans to retire at 70)

Now, we need to rearrange the formula to solve for P (the principal amount):

P = A / (1 + r/n)^(nt)

Plugging in the given values:
A = $130,000,
r = 0.14,
n = 2, and
t = 20,

Let's calculate the principal amount (P) that Ray must invest today:

P = $130,000 / (1 + 0.14/2)^(2*20)

Simplifying the equation, we have:

P = $130,000 / (1 + 0.07)^40
P = $130,000 / 1.07^40

Using a calculator, we find:

P ≈ $15,330.35

Therefore, Ray must invest approximately $15,330.35 today to meet his $130,000 retirement goal.